Flexible Robot
Manipulators
Other volumes in this series:
Volume 2 Elevator traffic analysis, design and control, 2nd editionG.C. Barney and S.M. dos Santos
Volume 8 A history of control engineering, 1800–1930S. Bennett Volume 14 Optimal relay and saturating control system synthesisE.P. Ryan Volume 18 Applied control theory, 2nd editionJ.R. Leigh
Volume 20 Design of modern control systemsD.J. Bell, P.A. Cook and N. Munro (Editors) Volume 28 Robots and automated manufactureJ. Billingsley (Editor)
Volume 30 Electromagnetic suspension: dynamics and controlP.K. Sinha Volume 32 Multivariable control for industrial applicationsJ. O’Reilly (Editor) Volume 33 Temperature measurement and controlJ.R. Leigh
Volume 34 Singular perturbation methodology in control systemsD.S. Naidu Volume 35 Implementation of self-tuning controllersK. Warwick (Editor)
Volume 37 Industrial digital control systems, 2nd editionK. Warwick and D. Rees (Editors) Volume 38 Parallel processing in controlP.J. Fleming (Editor)
Volume 39 Continuous time controller designR. Balasubramanian Volume 40 Deterministic control of uncertain systemsA.S.I. Zinober (Editor)
Volume 41 Computer control of real-time processesS. Bennett and G.S. Virk (Editors) Volume 42 Digital signal processing: principles, devices and applicationsN.B. Jones and
J.D.McK. Watson (Editors)
Volume 43 Trends in information technologyD.A. Linkens and R.I. Nicolson (Editors) Volume 44 Knowledge-based systems for industrial controlJ. McGhee, M.J. Grimble and
A. Mowforth (Editors)
Volume 47 A history of control engineering, 1930–1956S. Bennett
Volume 49 Polynomial methods in optimal control and filteringK.J. Hunt (Editor) Volume 50 Programming industrial control systems using IEC 1131-3R.W. Lewis
Volume 51 Advanced robotics and intelligent machinesJ.O. Gray and D.G. Caldwell (Editors) Volume 52 Adaptive prediction and predictive controlP.P. Kanjilal
Volume 53 Neural network applications in controlG.W. Irwin, K. Warwick and K.J. Hunt (Editors)
Volume 54 Control engineering solutions: a practical approachP. Albertos, R. Strietzel and N. Mort (Editors)
Volume 55 Genetic algorithms in engineering systemsA.M.S. Zalzala and P.J. Fleming (Editors) Volume 56 Symbolic methods in control system analysis and designN. Munro (Editor) Volume 57 Flight control systemsR.W. Pratt (Editor)
Volume 58 Power-plant control and instrumentationD. Lindsley Volume 59 Modelling control systems using IEC 61499R. Lewis
Volume 60 People in control: human factors in control room designJ. Noyes and M. Bransby (Editors)
Volume 61 Nonlinear predictive control: theory and practiceB. Kouvaritakis and M. Cannon (Editors)
Volume 62 Active sound and vibration controlM.O. Tokhi and S.M. Veres
Volume 63 Stepping motors: a guide to theory and practice, 4th editionP.P. Acarnley Volume 64 Control theory, 2nd editionJ. R. Leigh
Volume 65 Modelling and parameter estimation of dynamic systemsJ.R. Raol, G. Girija and J. Singh
Volume 66 Variable structure systems: from principles to implementationA. Sabanovic, L. Fridman and S. Spurgeon (Editors)
Volume 67 Motion vision: design of compact motion sensing solution for autonomous systems J. Kolodko and L. Vlacic
Volume 69 Unmanned marine vehiclesG. Roberts and R. Sutton (Editors)
Volume 70 Intelligent control systems using computational intelligence techniquesA. Ruano (Editor)
Flexible Robot
Manipulators
Modelling, simulation and control
Edited by
M.O. Tokhi and A.K.M. Azad
Published by The Institution of Engineering and Technology, London, United Kingdom © 2008 The Institution of Engineering and Technology
First published 2008
This publication is copyright under the Berne Convention and the Universal Copyright Convention. All rights reserved. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act, 1988, this publication may be reproduced, stored or transmitted, in any form or by any means, only with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Inquiries concerning reproduction outside those terms should be sent to the publishers at the undermentioned address:
The Institution of Engineering and Technology Michael Faraday House
Six Hills Way, Stevenage Herts, SG1 2AY, United Kingdom www.theiet.org
While the authors and the publishers believe that the information and guidance given in this work are correct, all parties must rely upon their own skill and judgement when making use of them. Neither the authors nor the publishers assume any liability to anyone for any loss or damage caused by any error or omission in the work, whether such error or omission is the result of negligence or any other cause. Any and all such liability is disclaimed.
The moral rights of the authors to be identified as authors of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.
British Library Cataloguing in Publication Data
Flexible robot manipulators: modelling, simulation and control. - (IET control series) 1. Manipulators (Mechanism) 2. Manipulators (Mechanism) - Automatic control I. Tokhi, M.O. II. Azad, Abul III. Institution of Engineering and Technology 629.8’92
ISBN 978-0-86341-448-0
Typeset in India by Newgen Imaging Systems (P) Ltd, Chennai Printed in the UK by Athenaeum Press Ltd, Gateshead, Tyne & Wear
Preface xv
List of contributors xvii
List of abbreviations xxi
List of notations xxv
1 Flexible manipulators – an overview 1
M.O. Tokhi, A.K.M. Azad, H.R. Pota and K. Senda
1.1 Introduction 1
1.2 Modelling and simulation techniques 2
1.3 Control techniques 4
1.3.1 Passive control 4
1.3.2 Open-loop control 5
1.3.3 Closed-loop control 5
1.3.4 Artificial intelligence control 9
1.4 Flexible manipulator systems 11
1.4.1 Typical FMSs 12
1.4.2 Flexible manipulators for industrial applications 13
1.4.3 Multi-link flexible manipulators 14
1.4.4 Two-link flexible manipulators 14
1.4.5 Single-link flexible manipulators 17
1.5 Applications 19
1.6 Summary 20
2 Modelling of a single-link flexible manipulator system: Theoretical
and practical investigations 23
A.K.M. Azad and M.O. Tokhi
2.1 Introduction 23
2.2 Dynamic equations of the system 25
2.2.1 The flexible manipulator system 25
2.2.2 Energies associated with the system 26
vi Flexible robot manipulators
2.3 Mode shapes 29
2.4 State-space model 30
2.5 Transfer function model 32
2.6 Experimentation 33 2.6.1 Natural frequencies 35 2.6.2 Damping ratios 39 2.6.3 Modal gain 43 2.7 Model validation 44 2.8 Summary 45
3 Classical mechanics approach of modelling multi-link flexible
manipulators 47
J. Sá da Costa, J.M. Martins and M.A. Botto
3.1 Introduction 47
3.2 Kinematics: the reference frames 48
3.2.1 Deformation assumptions 49
3.2.2 Kinematics of a flexible link 51
3.3 The strain–displacement relations 52
3.3.1 Parameterisation of the rotation matrix 55
3.3.2 Parameterisation of the neutral axis tangent vector 55
3.3.3 Displacement of the neutral axis 56
3.4 The dynamic model of a single flexible link 57
3.4.1 The inertial force term 57
3.4.2 The elastic force term 60
3.4.3 The gravitational force term 61
3.4.4 The external force term 61
3.4.5 Rayleigh–Ritz discretisation 62
3.5 The dynamic model of a multi-link manipulator 65
3.5.1 Joint kinematics 66
3.5.2 Dynamics of a rigid body 69
3.5.3 Dynamics of a rigid–flexible–rigid body 70
3.5.4 Dynamics of a serial multi-RFR body system 72
3.6 Summary 75
4 Parametric and non-parametric modelling of flexible
manipulators 77
M.H. Shaheed and M.O. Tokhi
4.1 Introduction 77
4.2 Parametric identification techniques 79
4.2.1 LMS algorithm 79
4.2.2 RLS algorithm 79
4.2.3 Genetic algorithms 80
4.3 Non-parametric identification techniques 81
4.3.1 Multi-layered perceptron neural networks 82
4.4 Model validation 85
4.5 Data pre-processing 87
4.6 Experimentation and results 87
4.6.1 Parametric modelling 87 4.6.2 Non-parametric modelling 90 4.6.2.1 Modelling with MLP NN 91 4.6.2.2 Modelling with RBF NN 94 4.7 Comparative assessment 95 4.8 Summary 96
5 Finite difference and finite element simulation of flexible
manipulators 99
A.K.M. Azad, M.O. Tokhi, Z. Mohamed, S. Mahil and H. Poerwanto
5.1 Introduction 99
5.2 The flexible manipulator system 100
5.3 The FD method 103
5.3.1 Development of the simulation algorithm 104
5.3.2 The hub displacement 105
5.3.3 The end-point displacement 105
5.3.4 Matrix formulation 106
5.3.5 State-space formulation 107
5.4 The FE/Lagrangian method 108
5.4.1 Elemental matrices 108
5.4.1.1 Scalar energy functions 109
5.4.2 A single-link flexible manipulator 110
5.4.3 A two-link flexible manipulator 111
5.4.3.1 Boundary conditions, payload and
damping 112
5.5 Validation of the FD and FE/Lagrangian methods 113
5.5.1 The experimental manipulator system 113
5.5.2 Simulation and experiments 113
5.6 Summary 117
6 Dynamic characterisation of flexible manipulators using symbolic
manipulation 119
Z. Mohamed, M.O. Tokhi and H.R. Pota
6.1 Introduction 119
6.2 FE approach to symbolic modelling 120
6.2.1 The flexible manipulator 121
6.2.2 Dynamic equation of motion 121
6.2.3 Transfer functions 124
6.2.4 Analysis 124
6.2.4.1 System without payload and hub inertia 125
6.2.4.2 System with payload 126
viii Flexible robot manipulators
6.3 Infinite-dimensional transfer functions using symbolic
methods 134
6.3.1 Piezoelectric laminate electromechanical
relationships 134
6.3.2 Dynamic modelling 136
6.3.3 Transfer functions 139
6.3.4 Rational Laplace domain transfer functions 141
6.3.5 Experimental system 142
6.3.6 Experimental results 145
6.4 Summary 146
7 Flexible space manipulators: Modelling, simulation, ground
validation and space operation 147
C. Lange, J.-C. Piedboeuf, M. Gu and J. Kövecses
7.1 Introduction 147
7.2 Symofros 149
7.2.1 Overview 150
7.2.2 Software architecture 151
7.2.3 Flexible beam modelling: a combined FE and
assumed-modes approach 152
7.3 Experimental validation 158
7.3.1 Experimental model validation using a single
flexible link 158
7.3.1.1 Experimental set-up 158
7.3.1.2 Simulation results 159
7.3.2 Flexible manipulator end-point detection and
validation 161
7.3.2.1 Flexible manipulator kinematics 162
7.3.2.2 Statics 165
7.3.2.3 End-point detection using strain gauges 168
7.4 SPDM task verification facility 178
7.4.1 Background 178
7.4.2 SPDM task verification facility concept 178
7.4.3 SPDM task verification facility test-bed 180
7.4.3.1 The SPDM task verification facility test-bed
simulator 180
7.4.3.2 The SPDM task verification facility test-bed
robot and robot controller 181
7.4.3.3 Computer architecture 183
7.4.3.4 ORUs and worksite 184
7.4.4 Experimental contact parameter estimation
using STVF 184
7.4.4.1 Description of the simulation
environment 185
7.5 On-orbit MSS training simulator 199
7.5.1 On-orbit training and simulation 201
7.5.2 Hardware architecture 201
7.5.3 Software architecture 201
7.5.4 Simulation validation 202
7.5.5 Symofros simulator engine 203
7.5.6 Analysis module 203
7.5.7 Ground and on-orbit results 203
7.6 Summary 206
7.7 Acknowledgements 206
8 Open-loop control of flexible manipulators using
command-generation techniques 207
A.K.M. Azad, M.H. Shaheed, Z. Mohamed, M.O. Tokhi and H. Poerwanto
8.1 Introduction 207
8.2 Identification of natural frequencies 208
8.2.1 Analytical approach 209
8.2.2 Experimental approach 209
8.2.3 Genetic modelling 211
8.2.4 Neural modelling 211
8.2.5 Natural frequencies from the genetic and neural
modelling 212
8.3 Gaussian shaped torque input 214
8.4 Shaped torque input 216
8.5 Filtered torque input 218
8.6 Experimentation and results 220
8.6.1 Unshaped bang-bang torque input 221
8.6.2 Shaped torque input 222
8.6.3 Gaussian shaped input 224
8.6.4 Filtered input torque 225
8.6.5 System with payload 228
8.7 Comparative performance assessment 230
8.8 Summary 233
9 Control of flexible manipulators with input shaping techniques 235
W.E. Singhose and W.P. Seering
9.1 Introduction 235
9.2 Command generation 238
9.2.1 Gantry crane example 238
9.2.2 Generating zero vibration commands 241
9.2.3 Using ZV impulse sequences to generate ZV
commands 244
x Flexible robot manipulators
9.2.5 Multi-mode input shaping 248
9.2.6 Real-time implementation 249
9.2.7 Trajectory following 250
9.2.8 Applications 250
9.3 Feedforward control action 252
9.3.1 Feedforward control of a simple system with
time delay 252
9.3.2 Zero phase error tracking control 255
9.4 ZPETC as command shaping 256
9.5 Summary 257
10 Enhanced PID-type classical control of flexible
manipulators 259
S.P. Goh and M.D. Brown
10.1 Introduction 259
10.2 Single-input single-output PI–PD 262
10.2.1 Basic algorithm 262
10.2.2 Discrete-time algorithm 262
10.3 Multi-input multi-output PI–PD 265
10.3.1 Basic notations 265 10.3.2 Decoupling algorithm 266 10.3.2.1 Strategy A 267 10.3.2.2 Strategy B 268 10.3.2.3 Strategy C 270 10.4 Experimental set-up 272
10.5 Simulation and experimental results 274
10.6 Summary 277
11 Force and position control of flexible manipulators 279
B. Siciliano and L. Villani
11.1 Introduction 279
11.2 Modelling 282
11.3 Indirect force and position regulation 286
11.3.1 First stage 286
11.3.2 Second stage 288
11.3.3 Simulation 288
11.4 Direct force and position control 292
11.4.1 Composite control strategy 292
11.4.2 Force and position regulation 294
11.4.3 Force regulation and position tracking 296
11.4.4 Simulation 297
12 Collocated and non-collocated control of flexible
manipulators 301
M.O. Tokhi, A.K.M. Azad, M.H. Shaheed and H. Poerwanto
12.1 Introduction 301
12.2 JBC control 303
12.2.1 Simulation results 304
12.2.2 Experimental results 304
12.3 Collocated and non-collocated feedback control involving PD
and PID 305
12.3.1 Simulation results 306
12.4 Adaptive JBC control 309
12.4.1 Simulation results 311
12.4.2 Experimental results 312
12.5 Adaptive collocated and non-collocated control 313
12.5.1 Simulation results 315
12.5.2 Experimental results 315
12.6 Collocated and non-collocated feedback control with PD and
neuro-inverse model 319
12.6.1 Simulation results 321
12.7 Summary 321
13 Decoupling control of flexible manipulators 325
G. Fernández, J.C. Grieco and M. Armada
13.1 Introduction 325
13.2 Multivariable control basics 326
13.3 Modelling a flexible link 328
13.3.1 Rigid–flexible robot case 328
13.3.2 Modelling the 2D flexible robot 329
13.4 Pre-compensator design 332
13.4.1 Rigid–flexible robot case 332
13.4.1.1 Column dominance for the rigid–flexible
robot 334
13.4.1.2 Column dominance for rigid–flexible robot
workspace 335
13.4.2 2D flexible robot case 335
13.4.2.1 Design of the decoupling filter for the 2D
flexible robot 335
13.5 Jacobian control of a 1D flexible manipulator 338
13.5.1 Jacobian control 339
13.5.2 Control results 341
xii Flexible robot manipulators
14 Modelling and control of space manipulators with flexible links 345
K. Senda
14.1 Introduction 345
14.2 Model of flexible manipulators 348
14.3 VRM concept 350
14.3.1 Definition of VRM 350
14.3.2 Kinematic relations of RFM and VRM 351
14.4 PD-control 355
14.4.1 PD-control for joint variables 355
14.4.2 Stability of linearised system 356
14.4.3 Stability of original non-linear system 357
14.5 Control using VRM concept 357
14.5.1 Control methods using the VRM concept 357
14.5.2 Asymptotic stability of positioning control 358
14.5.3 Stability of continuous path control 359
14.6 Control examples 361
14.6.1 Positioning control 361
14.6.2 Path control: hardware experiment 362
14.6.3 Composite control 363
14.7 Summary 364
14.8 Acknowledgement 365
15 Soft computing approaches for control of a flexible manipulator 367
S.K. Sharma, M.N.H. Siddique, M.O. Tokhi and G.W. Irwin
15.1 Introduction 367
15.2 The flexible manipulator system 369
15.3 Modular NN controller 370
15.3.1 Genetic representation of MNN architecture 371
15.3.1.1 Genetic encoding of NNs 371
15.3.1.2 Genetic learning for NN 372
15.3.2 Implementation and simulation results 374
15.3.2.1 End-point position tracking 374
15.3.2.2 Performance of MNN controller 375
15.4 FL control of a flexible-link manipulator 377
15.4.1 PD-type fuzzy logic control 377
15.4.2 PI-type fuzzy logic control 379
15.4.2.1 Integral wind-up action 381
15.4.3 PID-type fuzzy logic controller 382
15.4.3.1 PD–PI-type fuzzy controller 383
15.4.3.2 Experimental results 384
15.4.4 GA optimisation of fuzzy controller 387
15.4.4.1 Genetic representation for membership
functions 387
15.4.4.2 Experimental results 390
16 Modelling and control of smart material flexible manipulators 395
Z.P. Wang, S.S. Ge and T.H. Lee
16.1 Introduction 395
16.2 Dynamic modelling of a single-link smart material robot 398
16.2.1 AMM modelling 402
16.2.2 FE modelling 404
16.2.2.1 FE analysis 404
16.2.2.2 Dynamic equations 407
16.3 Model-free regulation of smart material robots 409
16.3.1 System description 409
16.3.2 Model-free controller design 410
16.3.2.1 Decentralised model-free control 410
16.3.2.2 Centralised model-free controller 411
16.4 Tracking control of smart material robots 422
16.4.1 Singular perturbed smart material robots 422
16.4.2 Adaptive composite controller design 425
16.4.2.1 Adaptive control of the slow subsystem 426
16.4.2.2 Stabilisation of fast subsystem 427
16.5 Summary 431
17 Modelling and control of rigid–flexible manipulators 433
A.S. Yigit
17.1 Introduction 433
17.2 Dynamic modelling 434
17.2.1 Discrete equations of motion 437
17.2.2 Convergence of the solution 439
17.3 Coupling between rigid and flexible motion 439
17.4 Impact response 441
17.5 Control of rigid–flexible manipulators 444
17.5.1 Stability of independent joint control for a two-link
rigid–flexible manipulator 446
17.5.2 Closed-loop simulations 447
17.6 Summary 450
18 Analysis and design environment for flexible manipulators 453
O. Ravn and N.K. Poulsen
18.1 Introduction 453
18.2 Computer aided control engineering design paradigm 455
18.3 Mechatronic Simulink library 458
18.4 Design models 460
18.4.1 Dynamics of actuators 461
18.4.2 Modal models 464
18.4.3 FE model 470
xiv Flexible robot manipulators
18.6 CACE environment 473
18.7 Summary 476
19 SCEFMAS – An environment for simulation and control of flexible
manipulator systems 477
M.O. Tokhi, A.K.M. Azad, M.H. Shaheed and H. Poerwanto
19.1 Introduction 477
19.2 The flexible manipulator system 478
19.3 Structure of SCEFMAS 479
19.3.1 FD simulation and control 481
19.3.1.1 FD simulation algorithm 482
19.3.1.2 Controller designs 482
19.3.2 Intelligent modelling and model validation 483
19.3.2.1 NN modelling 484
19.3.2.2 GA modelling 484
19.3.3 Graphical user interfaces 485
19.3.3.1 SCEFMAS_V2 GUI 486
19.3.3.2 Results GUI 486
19.3.3.3 NN modelling and validation GUIs 486
19.3.3.4 GA modelling and validation GUI 487
19.4 Case studies 487
19.4.1 Open-loop FD simulation 487
19.4.2 Adaptive inverse dynamic active control 492
19.4.3 NN modelling and validation 493
19.4.4 GA modelling and validation 495
19.5 Summary 498
References 501
The ever-increasing utilization of robotic manipulators in various applications in recent years has been motivated by the requirements and demands of industrial automation. Among the rigid and flexible manipulator types, attention is focused more towards flexible manipulators. This is owing to various advantages such manipulators offer as compared to their rigid counterparts. Exploitation of the potential benefits and capabilities of rigid and flexible manipulators introduces a further emerging line of research in which hybrid rigid–flexible manipulator structures are considered.
Flexural dynamics (vibration) in flexible manipulators has been the main research challenge in the modelling and control of such systems. Accordingly, research activ-ities in flexible manipulators have looked into the development of methodologies to cope with the flexural motion dynamics of such systems.
A considerable amount of research on the development of dynamic models of flexible manipulators has been carried out. These have led to descriptions in the form of either partial differential equations, or finite-dimensional ordinary differential equations. From a control perspective, an input/output characterisation of the system is desired, which can be obtained through suitable online estimation and adaptation mechanisms. Given the dynamic nature of flexible manipulator systems, the practical realisation of such methodologies presents new challenges.
Numerical techniques using finite difference and finite element methods have been researched for dynamic characterisation of flexible manipulators. Accordingly, simulation algorithms characterising the dynamic behaviour of flexible manipulators have been developed that provide flexible means of analysis, test and verification of control techniques. With the widely available use of digital computing technology, such platforms are first-step favoured option in a wide range of applications.
Control structures adopted for flexible manipulators can broadly be separated into open loop and closed loop. Although the mathematical theory of open loop control is well established, only a limited number of successful applications in the control of distributed parameter flexible manipulator systems have been reported. A further research dimension, with this class of control structures, is online adaptation of the input shaping mechanism with the changing behaviour of the system and the environment. With closed-loop control techniques, a common trend that has been adopted by researchers is partitioning of the dynamics of the system into the slow
xvi Flexible robot manipulators
(rigid-body) and fast (flexural motion) dynamics and accordingly devising separate control loops. An important consideration with this has been to adequately cope with the non-minimum phase behaviour exhibited by the system characterisation, which with optimal feedback control techniques leads to unstable control. Although this problem can be avoided with some traditional techniques, emerging intelligent control methodologies incorporating soft-computing paradigms offer a great deal of potential in solving such problems.
This book reports on recent and new developments in modelling, simulation and control of flexible robot manipulators, in light of the issues mentioned above. The contents of the book are divided into 19 chapters. Following a general overview of flexible manipulators from the perspective of modelling, simulation, control and applications in Chapter 1, the rest of the book may be grouped into four parts, although some overlap between the parts is allowed for reasons of completeness and coherency as far as required: (1) Chapters 2–4 provide a range of modelling approaches including classical techniques based on the Lagrange equation formula-tion and parametric approaches based on linear input–output models using system identification techniques and neuro-modelling approaches; (2) Chapters 5–7 present numerical modelling/simulation techniques for dynamic characterisation of flexible manipulators using the finite difference, finite element, symbolic manipulation and customised software techniques, with Chapter 7 dedicated to manipulators in space; (3) Chapters 8–17 present a range of open-loop and closed-loop control techniques based on classical and modern intelligent control methods including soft-computing and smart structures for flexible manipulators; (4) Chapters 18 and 19 are dedicated to software environments for analysis, design, simulation and control of flexible manipulators.
The book is intended for teaching in graduate courses on robotics, mechatronics, control, electrical and mechanical engineering. It can also serve as a source of refer-ence for research in areas of modelling, simulation and control of dynamic flexible structures in general and, specifically, of flexible robotic manipulators.
The material presented in this book comprises contributions of worldwide researchers in the field, and the editors are grateful for their professional and sci-entific support. The editors would also like to thank Professor Derek Atherton for his encouragement and support during the initial planning of this project, and the IET publication team for their patience and support throughout this project.
M.O. Tokhi
University of Sheffield, UK
A.K.M. Azad
M. Armada
Automatic Control Department Industrial Automation Institute Spanish National Research
Council (CSIC) Madrid, Spain
A.K.M. Azad
Department of Technology Northern Illinois University Illinois, USA
M.A. Botto
DEM–IDMEC
Instituto Superior Técnico Technical University of Lisbon Lisbon, Portugal
M.D. Brown
Lockheed Martin UK Integrated Systems
Havant, Hampshire, UK
G. Fernández
Electronic and Circuits Department Universidad Simón Bolívar Caracas, Venezuela
S.S. Ge
Department of Electrical and Computer Engineering National University of Singapore Singapore S.P. Goh GGS Systems (M) Sdn. Bhd. Johor Bahru Johor, Malaysia J.C. Grieco
Electronic and Circuits Department Universidad Simón Bolívar Caracas, Venezuela
M. Gu
Space Technologies Canadian Space Agency Québec, Canada
G.W. Irwin
School of Electrical and Electronics Engineering
Queen’s University of Belfast Belfast, UK
xviii Flexible robot manipulators
J. Kövecses
Mechanical Engineering Department McGill University
Montreal, Canada
C. Lange
Space Technologies Canadian Space Agency Québec, Canada
T.H. Lee
Department of Electrical and Computer Engineering National University of Singapore Singapore S. Mahil College of Engineering Purdue University Indiana, USA J.M. Martins DEM–IDMEC
Instituto Superior Técnico Technical University of Lisbon Lisbon, Portugal
Z. Mohamed
Faculty of Electrical Engineering University Technology Malaysia Malaysia
J.-C. Piedboeuf
Space Technologies Canadian Space Agency Québec, Canada
H. Poerwanto
PT PAL
Surabaya, Indonesia
H.R. Pota
Australian Defence Force Academy Canberra, Australia
N.K. Poulsen
Department of Informatics and Mathematical Modelling Technical University of Denmark Lyngby, Denmark
O. Ravn
Automation, Ørsted. DTU Technical University of Denmark Lyngby, Denmark
J. Sá da Costa
DEM–IDMEC
Instituto Superior Técnico Technical University of Lisbon Lisbon, Portugal
W.P. Seering
Department of Mechanical Engineering Massachusetts Institute of Technology Cambridge, USA
K. Senda
Division of Mechanical Science and Engineering
Kanazawa University Kanazawa, Japan
M.H. Shaheed
Department of Engineering Queen Mary and Westfield College University of London London, UK S.K. Sharma School of Engineering University of Plymouth Plymouth, UK B. Siciliano
Dipartimento di Informatica e Sistemistica Università degli Studi di Napoli
Federico II Napoli, Italy
M.N.H. Siddique
School of Computing and Intelligent Systems
The University of Ulster Londonderry, UK
W.E. Singhose
George W. Woodruff School of Mechanical Engineering Georgia Institute of Technology Atlanta, USA
M.O. Tokhi
Department of Automatic Control and Systems Engineering
The University of Sheffield Sheffield, UK
L. Villani
Dipartimento di Informatica e Sistemistica
Università degli Studi di Napoli Federico II
Napoli, Italy
Z.P. Wang
Department of Electrical and Computer Engineering National University of Singapore Singapore A.S. Yigit Department of Mechanical Engineering Kuwait University Safat, Kuwait
AB Articulated body
ACC Adaptive composite controller
ACU Arm computer unit
A/D Analogue/digital
ADAM Aerospace dual-arm flexible manipulator
AMM Assumed modes method
ANN Artificial neural network
ARMAX Autoregressive moving average with exogeneous inputs ARX Autoregressive with exogenous inputs
AVC Active vibration control
BC Boundary condition
CACE Computer aided control engineering
CCD Charge coupled device
CDT Contact dynamics toolkit
CLIK Closed-loop inverse kinematics CMFC Centralised model-free controller
CMM Coordinate measuring machine
CI Composite inertia
CRB Composite rigid body
CSA Canadian Space Agency
D/A Digital/analogue
DFM Duisburg flexible manipulator DMFC Decentralised model-free controller
DNA Direct Nyquist array
DOF Degree of freedom
DSP Digital signal processing
EAP Electroactive polymer
EBRC Energy-based robust controller ERLS Equivalent rigid link system EVA Extravehicular activity
FD Finite difference
xxii Flexible robot manipulators
FMA Force moment accommodation
FMS Flexible manipulator system
FRF Frequency response function
GA Genetic algorithm
GOCF Generalized observability canonical form
GUI Graphical user interface
HC Hard computing
HLS Hardware-in-the-loop simulation
IIR Infinite impulse response
IMSC Independent modal space control
INA Inverse Nyquist array
I/O Input/output
ISS International Space Station
ISTE Integral squared timed error
JBC Joint-based collocated
LED Light-emitting diode
LHP Left half of s-plane
LMS Least mean squares
LPDC Local proportional, derivative control
LQ Linear quadratic
LQG Linear quadratic Gaussian
LQR Linear quadratic regulator
LRMS Long-reach manipulator system
MAM Manual augmented mode
MBS Mobile base system
MDR MacDonald Dettwiler Space and Advanced Robotics Ltd. MDSF Manipulator development and simulation facility
MF Membership function
MIMO Multi-input multi-output
MIQ Machine intelligence quotient
MLFM Multi-link flexible manipulator
MLP Multi-layered perceptron
MNN Modular neural network
MOTS MSS operation and training simulator
MPIPD Multivariable PI–PD
MPO Model predicted output
MPP Multivariable pole-placement
MRO MSS robotics operator
MSL Mechatronic Simulink library
MSS Mobile servicing system
MTF Matrix transfer function
NARMAX Non-linear autoregressive moving average with exogeneous inputs NARX Non-linear autoregressive with exogeneous inputs
NB Negative big
NRT Non-real-time
NS Negative small
ODE Ordinary differential equation
OLS Orthogonal least squares
OPDE Ordinary partial differential equation
ORU Orbital replacement unit
OSA One-step-ahead
OTCM Orbital tool change-out mechanism
OTCME ORU tool change-out mechanism emulator
PB Positive big
PD Proportional, derivative
PDE Partial differential equation
PI Proportional, integral
PID Proportional, integral, derivative
PR Probabilistic reasoning
PRBS Pseudo-random binary sequence
PS Positive small
PSD Power spectral density
PZT Lead Zirconate Titanate (piezoelectric ceramic material)
RAC Resolved-acceleration control
RBF Radial basis function
RC Resistance–capacitance
RFM Real flexible manipulator
RFR Rigid–flexible–rigid
RH Routh–Hurtwiz
RHP Right half of s-plane
RLS Recursive least squares
RMRC Resolved-motion-rate control
RMS Remote manipulator system
RTAI Real-time application interface
RTW Real-time workshop
RVDT Rotary variable differential transformer
SC Soft computing
SCEFMAS Simulation and Control Environment for Flexible Manipulator Systems
SHM Shared memory
SIM Dynamic simulator
SISO Single-input single-output
SLFM Single-link flexible manipulator
SM Symbolic manipulation
SMG Symbolic model generator
SMP System for monitoring and maintaining MSS robotics operators performance
SMT STVF test-bed
xxiv Flexible robot manipulators
SPDM Special purpose dextrous manipulator SSRMS Space station remote manipulator system STVF SPDM test verification facility
TLFM Two-link flexible manipulator
USB Universal Serial Bus
VR Visual renderer
VRM Virtual rigid manipulator VSC Variable structure control
ZN Ziegler–Nichols
ZPETC Zero phase error tracking controller
ZV Zero vibration
ZVD Zero vibration and derivative
1D One-dimensional
1DOF One-degree-of-freedom
2D Two-dimensional
2DOF Two-degrees-of-freedom
a Thickness of beam
b Width of beam and of smart material
ai, bi, ci Constants, parameters, coefficients of polynomials
ap Absolute linear acceleration vector of point p expressed in the body reference frame
an,ak Absolute linear acceleration vector of the body reference frame and of the cross section reference frame, expressed in the body reference frame
aij Matrix element
A Cross-sectional area
An Generalized acceleration vector of body reference frame n
A, B, C System state matrices
bmj Bias on the jth neuron of the mth layer
Bno,Bno A body and its surface in the reference undeformed configuration B∈ R3 Magnetic flux density vector
cM ∈ R6×6 Symmetric matrix of elastic stiffness coefficients of the beam
cS∈ R6×6 Symmetric matrix of elastic stiffness coefficients of the smart material c1 Thickness of upper surface smart material patch
cM11 Stiffness of the beam
cS11 Stiffness of the piezoelectric material
c2 Thickness of lower surface smart material patch cL1 Stiffness per unit length of the beam
cL2 Stiffness per unit length of the smart material C, Cn Kinetic energy, capacitance
Ca Actuator voltage constant
Cs Sensor voltage constant
d Constant
di Components of the displacement gradient strain vector d31 Piezoelectric charge constant
D Damping matrix
xxvi Flexible robot manipulators
D(x, t) ∈ R3 Electrical displacement at location x and time t
e Error
˙e Change in error
e1, e2, e3 Unit vectors along the axis of the cross section reference frame, expressed in the body reference frame
E1, E2, E3 Unit vectors along the axis of the body reference frame, expressed in the inertial reference frame
E Young modulus
Ea Actuating layer Young’s modulus
EK System kinetic energy
EP System potential energy
E∈ R3 Electrical field intensity vector
E [.] Expectation
fn Natural frequency in Hz
fs Force per unit area applied on the surface of a beam
fn, fˆn Force per unit area applied on the base and tip of a beam
f ¯Bno Force per unit area applied on the lateral faces of a beam excluding the edges of the first and last cross section
F(t) Force function
Fn, Mn Resulting external force and moment applied at the origin of the body reference frame
F ¯Bno , M ¯B
no Resulting external force and moment per unit length applied at the
origin of the cross section reference frame
Fˆn,Mˆn Resulting external force and moment applied at the origin of the tip cross section reference frame
FM ∈ R6 Simplified stress vector of the beam
FS∈ R6 Simplified stress vector of the smart material
g Number of generation in genetic algorithms
g Gravitational acceleration vector expressed in the body reference frame
gmax Maximum number of generation in genetic algorithms
g31 Piezoelectric stress constant
G Green–Lagrange strain tensor
G Half Young modulus,G= E/2
Gb Green–Lagrange strain vector at a beam cross section
G(s) Transfer function (continuous)
h∈ R6×3 Coupling coefficients matrix
h12 Coupling parameter per unit volume of the piezoelectric material hL Coupling parameter per unit length of the smart material robot
H(x, t) ∈ R3 Magnetic field intensity at location x and time t
H Depth/height of arm/link
H(jω) Frequency response function
Hnω, Hnv Rotation and translation Jacobian matrices of joint n
I Area moment of inertia
I Identity matrix
Ih Hub inertia
Ip Inertia associated with payload
IT Total inertia
Ih+ l3ρA/3
j Constants, index, polynomial/model order, unit imaginary number
J Cost function
J kJ expressed in the body reference frame
J , I2, I3 Torsional and bending geometric moments of inertia.
JRek Elastic rotation Jacobian of the kth cross section
Jn Second moment of inertia tensor of rigid body n expressed in the body reference frame
JTk Elastic translation Jacobian of the kth cross section
k Constant, index
kJ Geometrical moments of inertia tensor of a cross section relative to and expressed in the cross section reference frame
Kc Control output scaling factor
K Stiffness matrix
Kk Vector of bending curvatures expressed in the cross section reference frame
Kn Elemental stiffness matrix
k31 Piezoelectric electromagnetic coupling constant
K1, K2, K3 Components of Kk. K1is the torsional strain, and K2and K3are the bending strains
Kp, Ki, Kd Proportional, integral, derivative parameters in PID control
Kv Derivative gain in PD control
l Elemental length L Length of arm/link Lg Lagrangian m Total number of NNs in an MNN m3 Payload at end-point M Mass matrix
Mn Elemental mass matrix
Men Mass matrix of flexible beam n
Ma(x, t) Local moment induced in beam by piezoelectric actuating layer
Mp Payload mass
mij Elements of mass matrix
n Constant, number of elements, number of modes
N Constant, number of samples
Nn Coriollis and centrifugal force terms of body n ¯
Nen Generalized force vector contemplating non-linear inertial forces,
linear elastic forces and the generalized external forces applied on the boundary of the beam
N(x), Na(x) Shape function vector
xxviiiFlexible robot manipulators {OI, XIYIZI} Inertial reference frame {On, XnYnZn} Body n reference frame
{Ok, XkYkZk} Beam cross section k reference frame
OX1Y1 Local reference frame with axis OX1tangential to the beam at the base
OX0Y0 Fixed base frame
p Constant, pole
Pc Crossover probability
Pm Mutation probability
Pcd Dynamic crossover probability
Pmd Dynamic mutation probability
P, Pn Potential energy
qen Global vector of the elastic generalized coordinates of a beam qKn Vector of pure bending displacement and pure torsion angle
generalized coordinates
qϕn Vector of pure shear displacement generalised coordinates qnω,qnv Vectors of angular and linear position parameters of joint n
Q(t), Qa(t) Nodal displacement vector
Q(t) Total charge in sensing layer
q(x, t) Charge distribution in sensing layer qi(t) Time-dependent generalized coordinates
r Position vector of point P expressed in the fixed base frame
r Reference input
rn Position vector of body reference frame n relative to and expressed in the inertial reference frame
rp Position vector of material point p relative to and expressed in the inertial reference frame
rn−1,n Vector describing the position of body n relative to body n− 1, expressed in body n− 1 reference frame
Rn Rotation matrix from the inertial reference frame to body n reference frame
Rek Rotation matrix from body n reference frame to cross section
k reference frame
Rn/n−1 Orthogonal rotation matrix expressing the rotation of body n relative to body n− 1
s Laplace variable
S∈ R6 Simplified strain vector
t Time (continuous)
t,kt Tangent vector to the beam neutral fibre expressed in the body reference frame, and expressed in the cross section reference frame ta, tb, tc Respective thicknesses of piezoelectric actuator, beam and
piezoelectric sensor
T Total elapsed time of the desired trajectory Tn Generalized control force vector at joint n
v(x, t) Voltage applied to the piezoelectric actuator Va(·) Actuator voltage
Vs(·) Sensor voltage
v System state
vp Absolute linear velocity vector of point p expressed in the body reference frame
vn, vk Absolute linear velocity vector of the
body reference frame and of the cross section reference frame, expressed in the body reference frame
V Generalized velocity vector of body reference frame n
w Elastic deflection
wijm Connection weight between the ith neuron of the(m − 1) th layer and the jth neuron of the mth layer
w(x, t) Deflection at location x and time t
W Width of arm/link, virtual work done by non-conservative forces
x Distance from hub along the arm/link
Xgn Centre of mass of rigid body n, relative to and expressed in the body
reference frame
X , Y , Z Moving coordinate system Xo, Yo, Zo Fixed coordinate system
Xp, xp, up Reference position, displaced position and displacement vector of a material point p, expressed in the body reference frame
Xk, xk, uk Reference position, displaced position and displacement vector of a material point on the beam neutral axis, expressed in the body reference frame
X1, X2, X3 Material coordinates of a body
y Plant output, total displacement, actual output
Yp Position vector of material point p in a given cross section, relative to and expressed in the cross section frame
ˆy Estimated/predicted output
z z-transform variable, zero
˜Z Skew symmetric matrix formed with the components of a given vector Z
z−1 Unit left-shift
α End-point acceleration, pure torsion angle
αk Absolute angular acceleration vector of a cross section expressed in the body reference frame
β Flexural rigidity
β ∈ R3×3 Symmetric matrix of impermittivity coefficients
β22 Impermittivity per unit volume of the piezoelectric material βL Impermittivity per unit length of the smart material robot δ Variational operator of the Principle of Virtual Powers
γi Pure shear deflections
k Strain vector of the beam neutral axis expressed in the cross section reference frame
xxx Flexible robot manipulators
1,2,3 Components ofk.1is the longitudinal strain, and2and3are the transverse Timoshenko shear strains
λ Number of correctly classified pattern
ψ Number of connections in the MNN
εc Strain in sensing layer
εij Components of the Green–Lagrange strain tensor φi(x) Admissible functions
φi Element of shape function vector
ϕ12,ϕ13 Shear angles as defined in the classical theory of elasticity Fn Generalized force vector applied at the origin of body reference
frame n
T
n+1,n Matrix transforming generalized forces applied at n+ 1 to generalized forces applied at n
θ, θ(t) Hub/joint angle
˙θ Hub/joint angular velocity
Joint angle at the hub
θd(t) Desired joint angular trajectory
ρ Mass density per unit volume
ρp Specific mass of a beam at point p
ρA Mass of a beam per unit length
ρ1 Mass per unit volume of the beam
ρ2 Mass per unit volume of the smart material
ρL1 Mass per unit length of the beam
ρL2 Mass per unit length of the smart material
σ Piola–Kirchoff stresss vector at a beam cross section δ Variational operator of the Principle of Virtual Powers
μ ∈ R3×3 Permeability coefficients matrix
μ33 Permeability of the piezoelectric material
μL Permeability per unit length of the smart material robot σa Longitudinal stress in actuating layer
τ Torque
τs Sample period
τ(t) Torque applied to the base of the manipulator
υ End-point residual
υi Pure bending deflections
υix,αx,γix Vectors of shape functions for the pure elastic deflections
υit ,αt,γit Vectors of the elastic generalised coordinates
ω Frequency in radian
ωc Cut-off frequency in radian
ωn Natural frequency in radian
ωn/n−1 Vector describing the angular velocity of body n relative to body n− 1 expressed in body n reference frame
ωn,ωk Absolute angular velocity vector of the body reference frame and of the cross section reference frame, expressed in the body reference frame
k Angular velocity vector of the cross section reference frame relative to the body reference frame, expressed in the body reference frame k Absolute angular acceleration matrix of a cross section expressed in
the body reference frame ζ , ζi Damping ratio
ξi Slope parameter of the activation function of ith NN output tanh(x) Activation function of hidden neurons
tanh(ξix) Activation function of ith NN output σ2
e Variance of variable e
Flexible manipulators – an overview
M.O. Tokhi, A.K.M. Azad, H.R. Pota and K. Senda
This chapter presents a general overview of previously developed methodologies for modelling, simulation and control of flexible manipulators. A selection of currently available flexible manipulator experimental systems in various research laboratories and outside laboratory environments are introduced and their features and design merits described. A structured overview of common applications and future research prospects and applications of flexible and hybrid manipulators are provided.
1.1
Introduction
Flexible manipulator systems (FMSs) offer several advantages in contrast to their traditional rigid counterparts. These include faster system response, lower energy consumption, the requirement of relatively smaller actuators, reduced non-linearity owing to elimination of gearing, lower overall mass and, in general, lower overall cost. However, owing to the distributed nature of the governing equations describ-ing dynamics of such systems, the control of flexible manipulators has traditionally involved complex processes (Aubrun, 1980; Book et al., 1986; Plunkell and Lee, 1970). Moreover, to compensate for flexural effects and thus yield robust control the design focuses primarily on non-collocated controllers (Cannon and Schmitz, 1984; Harashima and Ueshiba, 1986).
Research on FMSs ranges from a single-link manipulator rotating about a fixed axis (Hastings and Book, 1987) to three-dimensional multi-link arms (Nagathan and Soni, 1986). However, experimental work, in general, is almost exclusively limited to single-link manipulators. This is because of the complexity of multi-link manipu-lator systems, resulting from more degrees of freedom and the increased interactions between gross and deformed motions. It is important for control purposes to recognise the flexible nature of the manipulator system and to build a suitable mathemati-cal framework for modelling of the system. The use of dynamic models for FMSs
2 Flexible robot manipulators
is threefold: forward dynamics, inverse dynamics and controller design. Flexible manipulators are distributed parameter systems with rigid body as well as flexible movements. There are two physical limitations associated with the system:
1. The control torque can only be applied at the joint,
2. Only a finite number of sensors of bounded bandwidth can be used and at restricted locations along the length of the manipulator.
Such issues are considered in this chapter through a structured overview of techniques for modelling, dynamic simulation and control of flexible manipulators.
1.2
Modelling and simulation techniques
According to reported results, dynamic models of flexible manipulators are described either by partial differential equations (PDEs) or by finite-dimensional ordinary differ-ential equations (ODEs) through some kind of approximation. Owing to the principles used, various types of model of flexible manipulator have been developed (Kanoh et al., 1986). These can be classified as
• Lagrange’s equation and modal expansion (Ritz–Kantrovitch) • Lagrange’s equation and finite element (FE) method
• Euler–Newton equation and modal expansion • Euler–Newton equation and FE method
• Singular perturbation and frequency-domain techniques.
A commonly used approach for solving a PDE that represents the dynamics of a manipulator, sometimes referred to as the separation of variables method, is to utilize a representation of the PDE, obtained through a simplification process, by a finite set of ordinary differential equations. This model, however, does not always represent the fine details of the system (Hughes, 1987). A method in which the flexible manipulator is modelled as a massless spring with a lumped mass at one end and a lumped rotary inertia at the other end has previously been proposed (Feliú et al., 1992; Oosting and Dickerson, 1988). In practice, dynamic models are mostly formulated on the basis of considering forward and inverse dynamics. In this manner, consideration is given to computational efficiency, simplicity and accuracy of the model. Here, a means of predicting changes in the dynamics of the manipulator resulting from changing configurations and loading is proposed, where predictions of changes in mode shapes and frequencies can be made without the need to solve the full determinantal equation of the system.
An alternative to modelling the manipulator in the time domain is to use a method based on frequency domain analysis (Book and Majette, 1983; Yuan et al., 1989). This method develops a concise transfer matrix model using the Euler–Bernoulli beam equation for a uniform beam. The weakness of this method is that it makes no allowance for interaction between the gross motion and the flexible dynamics of the manipulator, nor can these effects be easily included in the model. As a result, the model can only be regarded as approximate. In another approach, a chain of flexible
links is modelled by considering a flexible multi-body dynamic approach, based on an equivalent rigid link system (ERLS), where an ERLS (which is the closest possible to the deformed linkage) is defined, in order to match, at best, the requirements of a small displacement assumption. As the choice of ERLS is completely arbitrary, it could introduce artificial kinematic constraints, which in turn introduce modelling error (Giovagnoni, 1994).
Unfortunately, the solutions obtained through the above modelling processes are approximate and do not represent fine details of a system. To resolve this problem, numerical solution of the system’s equation is performed allowing development of simulation environments. Dynamic simulation is important from a system design and evaluation viewpoint. It provides a characterisation of the system in the real sense as well as allowing online evaluation of controller designs. Commonly used simu-lation approaches involve finite element (FE), finite difference (FD) and symbolic manipulation (SM) methods. The FE method has been previously utilized to describe the flexible behaviour of manipulators (Dado and Soni, 1986; Usoro et al., 1984). The steps involved in FE simulation are discretisation of the structure into small ele-ments; selection of an approximating function to interpolate the result; derivation of an equation for these small elements; calculation of the system equation and solving the system equation considering the boundary conditions. The development of the algorithm can be divided into three main parts: the FE analysis, state-space represen-tation and obtaining and analysing the system transfer function. The compurepresen-tational complexity and consequent software coding involved in the FE method is a major disadvantage of this technique. However, as the FE method allows irregularities in the structure and mixed boundary conditions to be handled, the technique is found to be suitable in applications involving irregular structures.
In applications involving uniform structures, such as manipulator systems, the FD method is found to be more appropriate. Simulation studies of flexible beam systems have demonstrated the relative simplicity of the FD method (Kourmoulis, 1990). The FD method is used to obtain an efficient numerical means of solving the PDE by developing a finite-dimensional simulation of the FMS through a discretisation, both, in time and space (distance) coordinates. The algorithm allows inclusion of distributed actuator and sensor terms in the PDE and modification of boundary conditions. The development of such an algorithm for a FMS has previously been reported (Tokhi and Azad, 1995a; Tokhi et al., 1995). The algorithm thus developed has been implemented digitally and simulation results characterising the behaviour of the system under various loading conditions have been reported.
Investigations with symbolic manipulation have resulted in automated symbolic derivation of dynamic equations of motion of rigid and flexible manipulators utiliz-ing Lagrangian formulation and assumed mode methods (Cetinkunt and Ittop, 1992; De Luca et al., 1988; Lin and Lewis, 1994), Hamilton’s principle and non-linear integro-differential equations (Low and Vidyasagar, 1988) and FD approximations (Tzes et al., 1989). These methods have demonstrated that the approach has some advantages, such as allowing independent variation of flexure parameters. A study on utilizing the symbolic manipulation approach, for the modelling and analysis of a flexible manipulator using FE methods has also been reported (Mohammed and
4 Flexible robot manipulators
Tokhi, 2002). It is argued that the effect of payload on the manipulator is impor-tant for modelling and control purposes, as successful implementation of a flexible manipulator control is contingent upon achieving acceptable uniform performance in the presence of payload variations. The developed model has been verified by using an experimental rig to demonstrate the performance of the symbolic algorithm in modelling and analysis of a flexible manipulator.
1.3
Control techniques
The dynamic behaviour of a flexible manipulator may be considered as a combination of rigid-body and flexible dynamics. Accordingly, control strategies devised for such systems are to take account of both rigid-body motion and flexible motion control. The former corresponds to methods developed within the framework of conventional rigid manipulator control. The latter, on the other hand, corresponds to approaches developed within the framework of vibration control of flexible structures.
Vibration control techniques for flexible structures are generally classified into two categories: passive and active control (Tokhi and Veres, 2002). Active control utilizes the principle of wave interference. This is realised by artificially generating anti-source(s) (actuator(s)) to destructively interfere with the unwanted disturbances and thus result in reduction in the level of vibration. Active control of FMSs can in general be divided into two categories: open-loop and closed-loop control. Open-loop control involves altering the shape of actuator commands by considering the physical and vibration properties of the FMS. The approach may account for changes in the system after the control input is developed. Closed-loop control differs from open-loop control in that it uses measurements of the system state and change the actuator input accordingly to reduce the system response oscillation.
1.3.1
Passive control
Passive control utilizes the absorption property of matter and thus is realised by a fixed change in the physical parameters of the structure, for example, adding viscoelastic material to increase the damping properties of the flexible manipulator. It has been reported that the control of vibration of a flexible manipulator by passive means is not sufficient by itself to eliminate structural deflection (Book et al., 1986). On the other hand, if only active control is used then, owing to actuator and sensor dynamics, destabilisation of modes near the bandwidth of the actuator or sensor may result (Aubrun, 1980). To avoid such destabilisation a certain amount of passive damping will be required to be employed, thus using hybrid control, that is, a combination of active and passive control methods. Combined active/passive control strategies have been proposed previously where low-frequency modes of vibration are controlled by active means and the modes with frequencies just above the actively controlled modes are controlled by passive means (Plunkell and Lee, 1970).
Several methods of passive vibration control of FMSs have been developed over the years. These mainly include methods of implementation of a constrained vis-coelastic damping layer to provide an energy dissipation medium (Kerwin, 1959) and
the utilization of composite materials in the construction of a flexible manipulator to provide higher strength and stiffness-to-weight ratio and larger structural damping than a metallic flexible manipulator Aubrun, 1980; Choi et al., 1988; Thompson and Sung, 1986). Observations have shown that although passive damping provides a sharp increase in damping at higher frequency modes, the lower frequency modes still remain uncontrolled. Moreover, the addition of viscoelastic material and a con-straining layer leads to an increase in the size and dynamic load of the system (Tzou, 1988).
1.3.2
Open-loop control
Open-loop control methods have been considered in vibration control where the con-trol input is developed by considering the physical and vibrational properties of the FMS. Although, the mathematical theory of open-loop control is well established, only few successful applications in the control of distributed parameter systems, including flexible manipulator, have been reported (Dellman et al., 1956; Singh et al., 1989). The method involves the development of suitable forcing functions in order to reduce the vibration at resonance modes. The methods developed include shape command methods, the computed torque technique and bang-bang control. Shaped command methods attempt to develop forcing functions that minimise vibra-tions and the effect of parameters that affect the resonance modes (Aspinwall, 1980; Meckl and Seering, 1990; Swigert, 1980; Wang, 1986). Common problems of con-cern encountered in these methods include long move (response) time, instability owing to un-reduced modes and controller robustness in the case of a large change of the manipulator dynamics.
In the computed torque approach, depending on the detailed model of the system and desired output trajectory, the joint torque input is calculated using a model inversion process (Moulin and Bayo, 1991). The technique suffers from several prob-lems, owing to, for instance, model inaccuracy, uncertainty over implementability of the desired trajectory, sensitivity to system parameter variations and response time penalties for a causal input.
Bang-bang control involves the utilization of single and multiple switched bang-bang control functions (Onsay and Akay, 1991). Bang-bang-bang control functions require accurate selection of switching time, depending on the representative dynamic model of the system. A minor modelling error could cause switching error and thus result in a substantial increase in the residual vibrations. Although, utilization of minimum energy inputs has been shown to eliminate the problem of switching times that arise in the bang-bang input (Jayasuriya and Choura, 1991), the total response time, however, becomes longer (Meckl and Seering, 1990; Onsay and Akay, 1991).
1.3.3
Closed-loop control
Effective control of a system always depends on accurate real-time monitoring and the corresponding control effort. Initial discussions on feedback control of a flexible manipulator and the usefulness of optimal regulator as applied to this problem date
6 Flexible robot manipulators
back to the early 1970s. It is known in the conventional approach that compensation can alter the first vibrational mode by either adding some damping or extending the bandwidth of the system (Ogata, 2001). Compensation, however, will limit the performance of the manipulator because inputs with frequency contents above the first flexible mode could still cause vibration. Various modern control designs have been proposed during the last two decades for FMSs with different types of vibration measuring systems.
When free motion of a system consists mainly of a limited number of clearly separable modes, then it is possible to control these modes directly using the so-called independent modal space control (IMSC) method, where the controller is designed for each mode independent of other modes (Baz et al., 1992; Sinha and Kao, 1991). Modal space control has been used for suppression of flexible motion in a three-link log loading manipulator with which considerable improvement has been achieved over conventional joint-based collocated controller. Although, initial investigations on the use of IMSC lack consideration of the location of the actuator (Meirovitch et al., 1983), later investigations have shown that actuator placement is important for suppression of spillover and, thus, methods for optimal placement of sensors and actuators have been developed (Schulz and Heimbold, 1983).
Variable structure control (VSC) utilizes a viable high-speed switching feedback control law to drive the plant’s state trajectory onto a specified and user-specified surface in the state-space, and to maintain the plant’s state trajectory on this surface for all subsequent times. One of the first studies on the application of VSC to one-link flexible manipulators was reported by Qian and Ma (1992), where they controlled the end-point position in a non-collocated manner. In this study, a sliding surface (a line in the study) is constructed from the end-point position and its derivative is employed in the design. Qian and Ma (1992) claim that if the slope of this line is chosen positive and the system variables are made to stay on this line, these would converge to zero exponentially, thus yielding a stable system in sliding mode. The performance of the controller was evaluated through a series of simulations, followed by an analysis of the designed control system. Thomas and Bandyopadhyay (1997), however, have pointed out that the choice of a positive constant as the slope for this switching line would not guarantee the stability of the system in sliding mode. The switching line is in fact a switching hyper-surface in view of the functional relationship of the tip (end-point) position with the generalized coordinates of the system through mode shape functions. The stability of the system in sliding mode is guaranteed only if the motion on this hyper-surface is asymptotically stable (Young, 1977), whereas the positive value for the slope of the switching line employed by Qian and Ma (1992) will not guarantee this stability. Moreover, a stable VSC controller based on a state transformation has been designed in this study.
The application of VSC to multi-link flexible manipulators is very limited. There are difficulties in both modelling and controller design. Sira-Ramirez et al. (1992) have derived dynamical sliding mode regulators within the context of generalized observability canonical form (GOCF) (Fliess, 1989). The GOCF is obtained by means of a state elimination procedure, carried out on the system of differential equa-tions describing the manipulator dynamics. Therefore the system can be considered